Businessman David Gonski, with Simon Birmingham and Malcolm Turnbull at Ermington West Public School in Sydney in April 30, 2018. Photograph: Mick Tsikas/AAP

Birmingham now concedes, for the first time, that it was a campaign that cost him his job. The price of peace, he says, was his own head on a platter.

“I was confident we could agree to a principled implementation ... with the nine in 10 Catholic education leaders who had always engaged with me in good faith,’’ Birmingham told Guardian Australia.

“But if we also wanted peace with the one in 10, then a new face was clearly a prerequisite. I was happy to suggest a number of credible options, including Dan [Tehan].”

They had their priests preaching from the pulpit on Sunday about how terrible the NSW government was

Adrian Piccoli

The “one in 10”, Birmingham says, is the Victorian Catholic Education Commission’s Stephen Elder, a former Victorian Liberal MP. Recently, Elder expressed his “regret” for Birmingham’s “humiliation” while appearing to claim responsibility for his scalp.

Birmingham was a rare breed in Turnbull’s cabinet – out of 18 Liberal MPs, he was one of only four who attended public schools.

Tasked with the difficult job of securing Senate support for the Gonski 2.0 package, which would deliver more cash than Abbott but less than promised by Gillard, it was decided the best plan was to hang it on principle – the Senate lure was an end to special deals and one funding system for all schools, public and private.

Senior Liberals now concede the one funding model agenda was crafted to appeal to the Greens, but it set the Turnbull government on a collision course with Catholics.

Instead of welcoming the new cash splash – there would be billions of dollars more for public, private and Catholic schools too – the Catholic sector was furious.

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Bishop Peter A Comensoli, who now serves as the archbishop of Melbourne, was among the first on the phone to express his displeasure. Guardian Australia has confirmed that he rang Birmingham, accusing him of blindsiding Catholic leaders.

After being duchessed by successive governments, the Gonski 2.0 announcement was a shock. But soon it became apparent that it was not simply the lack of consultation – or the threat to funding – that was the problem. It was the government’s plan to publish an online estimator of schools funding and to write to parents how much their school would get that made Catholic educators furious.

“There were multiple representations, including from the now archbishop of Melbourne, over that intervening period about the fact that we were going to tell every school what they were going to get, urging us not to do that,’’ a senior Liberal confirms.

As Comensoli explained to the minister, the taxpayer-funded grants for each school went into one big “pot” to be redistributed. The letter from the commonwealth to parents and the actual funds schools received would not tally. (The archbishop declined requests to discuss this opposition to the transparency on how taxpayer funds were spent, citing his busy schedule at the Vatican.)

Two federal byelections were used to press the Catholics’ case – Batman in Victoria won by the Labor party, and Longman in Queensland.

Catholic Education conducted robocalls in Batman campaigning for Labor, following party leader Bill Shorten’s confirmation during the campaign that Catholic schools would secure $250m more over two years. The church also wrote to parents before the byelections to warn of possible fee rises and school closures.

But many Labor observers dismiss the Catholic sector’s claim of powerful interventions in the context as “a beat up”. They say Shorten’s contact with Catholics to thank them for the robocalls was nothing more than stakeholder management.

Despite this, the ALP’s Victorian deputy state director, Kos Samaras, believes the idea of slashing Catholic school funding would be politically disastrous.

“Aside from the myriad of other vote killing policies that we have embraced in the past, going down this path will wipe Labor out,’’ he wrote in a recent Facebook post to Labor supporters.

“It would cost billions to make up the shortfall … around 200,000 odd Labor primary votes.”

Adrian Piccoli, with New South Wales premier Mike Baird (right). The former NSW education minister, a believer who sends his children to a Catholic school, says politicians are running scared. Photograph: Paul Miller/AAP

The true purpose of the fund, according to insiders, was to deliver the cash to the Catholic sector that the parliament would not countenance. It could be done by regulation and did not require legislation. Government sources confirm it was the brainchild of independent schools.

The former New South Wales education minister Adrian Piccoli, a believer who sends his children to a Catholic school, says politicians are running scared.

We were never dishonest. We never lied

Ross Fox

During his tenure, Piccoli – who remains incensed that the Catholic church is resisting calls to report child sex offenders, citing the sanctity of the confessional – learned the hard way when he tried to cut funding.

“They had their priests preaching from the pulpit on Sunday about how terrible the NSW government was because we were going to take a relatively small amount of money off them.

“And you know what? It really spooks MPs and I just don’t know why. I don’t actually think that Catholics, and I am one of them, have that much faith in the Catholic hierarchy.’’

The Grattan Institute’s Peter Goss maintains that as few as 30 Catholic primary schools faced fee increases in excess of $4,000 a year and the median family income at those schools was $200,000.

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But Ross Fox, the director of Catholic education for Canberra and Goulburn – hardest hit by the changes – says their complaints about the unfairness of the original needs-based funding model had been vindicated by the change to a new model using parents’ tax data.

“We had a responsibility to speak. At various points people used the words ‘dishonest’ and ‘liars’. Pyne certainly said ‘dishonest’. I take personal offence to that because we were never dishonest. We never lied,’’ he says.

“Birmingham’s mistake was to think that policy purity was going to serve the interests of school children in Australia. He thought the most important thing was to get one funding model for every single school and so he put that ahead of whether schools were going to close, and fees,’’ Fox says.

The Greens leader, Richard Di Natale, has accused the government of pandering to “bullies” in the sector and having “sold Australian students down the river to silence a powerful lobby group”.

Labor’s own endgame was always to outspend the Coalition, particularly on public schools. And more money for public schools is a battle the Liberals cannot win.